Chapter 4 A Development of Novel Integral Method for Prediction of Distorted Inlet Flow Propagation in Axial Compressor Based on the original integral method and its applications mentioned and discussed in previous chapters, an improved integral method is proposed and developed for the quantitative prediction of distorted inlet flow propagation through axial compressor. The novel integral method is formulated using more appropriate and practical airfoil characteristics, with less assumption needed for derivation. The results indicate that the original integral method [6] underestimated the propagation of inlet flow distortion. The effects of inlet flow parameters on the propagation of inlet distortions, as well as on the compressor performance and characteristic are simulated and analyzed. From the viewpoint of compressor ef- ficiency, the propagation of inlet flow distortion is further described using a compressor critical performance and its associated critical characteristic. The re- sults present a realistic physical insight to an axial flow compressor behavior with a propagation of inlet distortion. 4.1 Introduction Compression system is an important component of the gas turbine engine, and its performance strongly influences the performance of all other components. The operational envelopes of modern compressors also demand a challenging trade-off between safety and performance due to the inherent aerodynamic in- stabilities associated with compressor stall. Various dynamic events such as dis- torted inlet flow in axial compressor, rotor blade tip clearance changes, rapid changes in the operating conditions (fuel throttling) may cause the component- mismatch and instability problems including rotating stall or even surge (oscil- lations in the mass flow rate), or a combination of them, which might result in catastrophic damage to the entire engine. Engines are thus constrained to operate below the surge line termed as a safe surge margin. Because the safe surge mar- gin must accommodate the most extreme events, the surge margin is therefore sizable and causes the compressor to operate below the optimal conditions. The ability to detect and prevent an impending stall allows the engine to be designed and safely operated at maximum efficiency without sacrificing engine weight in 78 future gas turbine engine development. The operability of current and future gas turbine engines can then be greatly enhanced. To understand and avoid compressor instability due to flow distortion, the inlet distortion and its propagation effects have received much attention over the years. The early work about analytical and experimental descriptions of propagating stall can be traced back to mid-fifties. In this period, one of remarkable work was done by Marble [8]. Marble proposed a simple model to yield essential features of stall propagation, such as dependence on the extent of stalled region upon operating conditions, the pressure loss associated with stall, and the angular velocity of stall propagation. In the work done by Emmons et al. [3], an experimental investigation was performed to verify their theory of instability about the phenomena in surge and stall propagation. More work had been done in the recent years. Cumpsty and Greitzer [2] proposed a simple model for compressor stall cell propagation. Jon- navithula et al. [5] presented a numerical and experimental study of stall propaga- tion in axial compressors. Longley et al. [7] described stability of flow through multistage axial compressors, and so on. As to the work focused on the analyses of distorted inlet flow, Reid [13] pre- sented an insight into the mechanism of the compressor’s response and tolerance to distortion. One of the methods used in the analyses was a linearized approach ([12] and [4]), that provides a quantitative information about the performance of the com- pressor in a circumferentially non-uniform flow. Several models ([10] and [4]), such as the parallel compressor model and its extensions [9], were used to assess the compressor stability with inlet distortion. Stenning [15] also presented some simpler techniques for analyzing the effects of circumferential inlet distortion. The numerical simulation of complex flows within multiple stages of turbo- machinery is becoming more effective and is useful for design application today by using the advanced computers. However, a large-scale simulation with CFD codes still requires huge computing resources far exceeding the practical limits of most single-processor supercomputers. Many CFD codes have to be performed on a parallel supercomputer ([1] and [17]). In order to rapidly predict the distorted performance and distortion attenuation of an axial compressor without using com- prehensive CFD codes and parallel supercomputer, it is necessary to make some simplifications, and some elegance and detail of flow physics must be sacrificed. Kim et al. [6] successfully calculated the qualitative trend of distorted perform- ance and distortion attenuation of an axial compressor by using an overly simpli- fied integral method. Instead of solving a detailed flow field problem, the integral method renders the multistage analysis as a nature part, and permits large velocity variations, including back flow. Ng et al. [11] developed the integral method and proposed a distortion critical line. The integral method provided a useful physical insight about the performance of the axial compressor with an inlet flow distor- tion. It is thus meaningful to further develop and refine this method. In the present study, the authors further improve and develop the integral method from the previous one [6] by adopting more appropriate and realistic air- foil characteristics. The calculated results indicate that the previous integral method underestimated greatly the inlet distortion propagation. By using the newly developed integral method, an investigation is proceeded to present the Chapter 4 A Development of Novel Integral Method effects of inlet parameters upon the downstream flow features with inlet distortion, including the inlet distortion propagation, the compressor critical performance and critical characteristic. The airfoil characteristics are also derived and discussed. distorted inlet f lo w undistorted inlet f lo w R π 2 R x Y y π 2)( 1 + = )( 2 x Y y = ) ] x (Y ) x (Y[5.0 ) x (Y y 21 + = = )( 1 x Y y = )( x δ )( x δ Fig. 4.1. The distorted inlet flow and its two-dimensional schematic used for inte- gral method 4.2 Theoretical Formulation Consider a two-dimensional inviscid flow through an axial compressor as shown in Fig. 4.1. In the x, y plane, the circumferential extent of compressor is denoted by y –direction and its period is 2πR. The flow field is divided as two parts, one is undistorted flow and another is the distorted flow. The distorted flow moves along a line )x(Yy = and extends a distance )x( δ on each side of this line to form a stream tube. Thus, the distorted region in y-direction is from )x()x(Yy δ − = to )x()x(Yy δ += , and the undistorted region is from )x()x(Yy δ + = to )x()x(YR2y δ π −+= . The flow that is subjected to a force field can be described by the equations of continuity and motion: 0 y v x u = ∂ ∂ + ∂ ∂ (4.1a) x 2 F) p ( xy )uv( x )u( = ∂ ∂ + ∂ ∂ + ∂ ∂ ρ (4.1b) y 2 F) p ( yy )v( x )uv( = ∂ ∂ + ∂ ∂ + ∂ ∂ ρ (4.1c) 4.2 Theoretical Formulation 79 distorted inlet undistorted inlet v , u , x 1 1 2 R π η = 1 2 R π η = − 1 η = 0 η =( / δ ) − 1 / δ η Fig. 4.2. A schematic of coordinates transformation on computational domain The integral technique is to integrate (4.1), with respect to y, and to determine the development of the flow from the inlet toward the downstream. Before integrating the equations of motion and continuity in the distorted re- gion, undistorted region and overall region respectively, the matching of velocity profiles and pressure field should be chosen. To illustrate the procedure of integral method, let consider a simple example. The velocities in each of the regions are taken to be independent of y whereas the velocity components in each of the regions are defined as: 0 U)x(u α = (4.2a) 0 V)x(v β = (4.2b) 000 U)x(u α = (4.2c) 000 V)x(v β = (4.2d) The inlet velocity has an angle of 0 θ , and: 000 tan V U γ θ == (4.3) where 0 U and 0 V are the x- and y- components of reference inlet velocity respec- tively. The distorted velocity coefficients )x( α and )x( β are the velocity fractions of the referenced inlet velocity in the distorted inlet region, and the undistorted ve- locity coefficients )x( 0 α and )x( 0 β are the velocity fractions of the referenced inlet velocity in the undistorted inlet region respectively. u and v are the x- and y- components of distorted velocity, while 0 u and 0 v are the x- and y- compo- nents of undistorted velocity respectively. flow Chapter 4 A Development of Novel Integral Method 80 For simplifying the derivation of integral equations, the coordinates system is transformed in the circumferential direction from (x, y) to (x, η) using )x( )x(Yy δ η − = . The computational domain is thus transformed into a parallel chan- nel, as shown in Fig. 4.2. 81 The static pressure is taken circumferentially (vertically) uniform: )x( pp ρρ ≡ (4.4) This assumption is taken in simplifying the process of derivation, which means that fluid pressure in the distorted region is fixed along y-direction, and this as- sumption would cause an overestimation of distortion level and propagation. In other words, if the decrease of static pressure is neglected, the inlet-distorted ve- locity that is calculated from the measured-inlet total pressure will be smaller than the reality. Therefore, the difference of the velocities between two regions at inlet will be increased compare with the reality, and the inlet distortion would be over- estimated. The predicted results of propagation of distortion would be larger than the real one, and hence provide a wider safety margin. The more the compressor’s stage number (longer axial scale) is, the higher would be the magnitude of overes- timated distortion. F θ v u F ⊥ F // F ⊥ F ⊥ ⊥ F // F // F // λ s λ r R OTO R ST A TO R θ r s u ω R -v Fig. 4.3. The force diagram and velocities in the compressor stage In the equations of motion and continuity, the force components x F and y F are the replacements of the equivalent terms acting on the blades as shown in Fig. 4.3. The multi-stages therefore become a natural part of compressor. The following are the definitions of the forces acting on the stator and rotor in the region of 4.2 Theoretical Formulation distortion according to the blade element theory (replacing u and v by u 0 and v 0 in the undistorted region): )vu( 2 C F 22 l s += ⊥ (4.5a) )vu( 2 C F 22 d s // += (4.5b) ])vR(u[ 2 C F 22 l r −+= ⊥ ω (4.5c) ])vR(u[ 2 C F 22 d r // −+= ω (4.5d) Here, l C and d C are the lift and drag coefficients respectively. The force compo- nents for a unit circumferential distance of a complete stage are: ( ) ( ) s s //s s s r r //r r r x cosFsinFcosFsinFF θθ λ λ θθ λ λ −+−= ⊥⊥ (4.6a) ( ) ( ) s s //s s s r r //r r r y sinFcosFsinFcosFF θθ λ λ θθ λ λ −−++= ⊥⊥ (4.6b) where the superscript and subscript s denote the stator; the superscript and sub- script r denote the rotor; the λλ r and λλ s are the relative length of rotor and stator in a single stage respectively. Here, we assume that sr λλλ += . The angles s θ and r θ are the local flow angles with respect to stator and rotor. For ex- ample, in distorted region, uvtan s = θ (4.7a) ( ) uvRtan r −= ω θ (4.7b) The ( v,u ) will be replaced by ( 00 v,u ) in undistorted region. With substitution of (4.2), (4.3), (4.5), (4.7) into (4.6), and simplifying the re- sulted equations by using 0 V R ω σ = , we obtain: 222 222 x s r ld ld 222 22 0 F [C ( ) C ] ( ) (C C ) U2 2 λ λ σ βγ α α σ β γ βγ α α βγ λσ γ λσ γ =−−+−+−+ (4.8a) y 222 222 s r ld ld 222 22 0 F [C C ( ) ] ( ) (C C ) U2 2 λ λ α σ βγ ασ βγ α βγ α βγ λσ γ λσ γ =+−+−−++ (4.8b) Chapter 4 A Development of Novel Integral Method 82 83 Similarly, the force components in the undistorted region 0,x F and 0,y F can be ob- tained using expression of undistorted velocity coefficients 0 α and 0 β . Here, the equations for the force components are different from the previous expressions [6]. The current effort is derived by adopting an exact blade element theory. On the other hand, unlike the oversimplified procedure with the constants lift and drag coefficients, the more practical coefficients are to be applied according to the ex- perimental results of airfoil sections [16]. In the distorted region, because the boundaries are streamlines, we obtain the following equations by integrating (4.1) along η-direction: ttanconsud 1 1 = ∫ − ηδ (4.9a) [] { } ∫∫∫ −−− = ∂ ∂ ′ + ′ −+ 1 1 x 1 1 1 1 2 dFd p )Y( dx d du dx d ηδη ρη ηδδηδ (4.9b) 11 11 ( , 1) ( , 1) y dpp uv d x x F d dx δ ηδη ρρ −− ⎡⎤ +−−= ⎢⎥ ⎣⎦ ∫∫ (4.9c) From (4.9a), the product of δα is a constant. For ease in derivation, we take: 1 K R ≡ π δα (4.10) From (4.9b) and (4.9c), we obtain: 2 0 x 0 U F ) p ( dx d U 1 dx d =+ ρ α α (4.11) and y 2 0 F d1 () dx U β α γ = (4.12) In the undistorted region, by integrating the equations of motion in [1, 1 R2 − δ π ], we thus obtain: 2 0 0,x 2 0 0 0 U F ) p ( dx d U 1 dx d =+ ρ α α (4.13) and: 4.2 Theoretical Formulation y,0 0 0 2 0 F d 1 () dx U β α γ = (4.14) Next, with integration of the continuity equation over the whole region: contant 2R 11 000 11 Ud Ud π δ δα ηδ α η − − +≡ ∫∫ (4.15) yields: 00000 K)RU2()]U)(2R2(U2[ ≡−+ παδπαδ (4.16) Using (4.10): 0011 K)K1(K =−+ αα (4.17) Rearranged as: 1 10 0 K )KK( − − = α α α (4.18) By differentiating the above equation will result in: d x d K d x d 2 0 α α = (4.19) where: 2 1 011 2 )K( )KK(K K − − = α (4.20) The integrated result of equation of motion in the overall region is equal to the combined results in both distorted and undistorted regions. In x-direction, by combining the (4.11) and (4.13), we obtain: 2 0 0,xx 0 0 U FF dx d dx d − =− α α α α (4.21) Substituting of (4.18) and (4.19) into (4.21), yields: 2 0 0,xx 3 U FF K 1 dx d − = α α (4.22) where: Chapter 4 A Development of Novel Integral Method 84 85 1 012 3 K )KK(K 1K − − += α (4.23) From the above results, we can rearrange the five ordinary differential equations, (4.11), (4.12), (4.14), (4.19), and (4.22) as follows: ) U FF ( K 1 dx d 2 0 0,xx 3 − = α α (4.24a) y 2 0 F d1 () dx U β α γ = (4.24b) dx d K dx d 2 0 α α = (4.24c) y,0 0 0 2 0 F d 1 () dx U β α γ = (4.24d) ) dx d U F (U) p ( dx d 2 0 x 2 0 α α ρ −= (4.24e) The above integral equations include five variables. They are two distorted veloc- ity coefficients )x( α and )x( β , two undistorted velocity coefficients )x( 0 α and )x( 0 β , and one static pressure (p/ρ). These integral equations can describe the development of both distorted and undistorted regions, as well as (and most im- portant) the progression of pressure in the compressor. The relative magnitude of vertical extension of distorted region is: )x(K R )x( )x( 1 α π δ ξ == (4.25) With calculation of the four velocity coefficients in solving the integral equations, the size of distorted region, )x( ξ , can then be computed using (4.25). There are two significant improvements between the current integral equation and the previous attempt. One of them is the force expression. In the previous papers, for simplicity, the lift and drag coefficients were assumed as constants and then an arti- ficial term including the angle of attack was inserted to correct the error induced by the assumption. This simplification resulted in the vanishing of lift and drag forces simultaneously at some flow angles, which is in a real flow unrealistic (non-physical). We thus employ actual airfoil characteristics in expressing the force components without any significant assumption, which should produce flow with better physics. The second contribution is in the derivation of integral equation. Kim et al. [6] used an equation with conservative form in distorted region but non-conservative form in undistorted region. We derive the equations in strong conservative form for both regions. This permits different results with the changes of integral equation, 4.2 Theoretical Formulation (4.24d), the inclusion of variation in pressure-density ratio, (4.24e), and the asso- ciated parametric expression, (4.23). These equations are different from the previ- ous effort. 4.3 Results and Discussion The integral equation has been coded and solved in Chap. 1, and the results were compared with that of Kim et al. [6]. Based on these research work, the current in- tegral equation is solved by using fourth order Runge-Kutta method with the fol- lowing initial conditions: λ λ λ 5.0 sr = = , 0.1)0()0( 00 = = β α , and )0()0( β α = . From the experimental test cases [14], the rotor blade speed is s/m6.36R = ω , 2= σ , and the airfoil blade section is NACA 65-series. 4.3.1 Lift and Drag Coefficients The most significant part in the current effort is the development of the application with airfoil characteristics. In the first step, we release/avoid the assumption of constant lift and drag coef- ficients, as well as the need of correcting terms in force components. Then we col- lated the necessary experimental data of both lift and drag coefficients. Finally, by curve fitting procedure, two sets of data are summarized as two cure formulas suitable to be employed as the expression of force components. The experimental data of NACA 65-series airfoil [16] indicates that the lift co- efficient has a linear relationship with the angle of attack in a normal range: ]10,8[ °°− . While the drag coefficient is the second or higher power of the lift coefficient, and there is a much smaller drag coefficient with a small angle of at- tack: ]2,1[ °°− . For example, a NACA 65-209 wing section with a lower Rey- nolds number: Re 6 3.0 10=× , the data can be described as: l C 0.1062 0.15 α =+ ( 810 α −°≤ ≤ ° ) (4.26a) 5 l 2 4 l 3 3 l 2 2 l 2 l 32 d C1044918.0C1020416.0C1060949.0 C1079495.0C1096313.0106518.0C −−− −−− ×+×−× −×+×−×= ( 1& 2 α α < −° > ° ) (4.26b) 0043.0C d = ( 12 α −°≤ ≤ ° ) (4.26c) Here, α is the wing section angle of attack. In the present case, (4.26a) is ob- tained by linear fitting, and the (4.26b) is calculated by Chebyshev curve fitting. Using (4.26) in (4.8), the force components can thus be obtained. Chapter 4 A Development of Novel Integral Method 86 [...]... 0.865,0.880,0.900,0.925,0.955,0.960,0.995 ,1. 040 ,1. 080 ,1. 10 To construct a Chebyshev multinomial : P( x ) = a1 + a 2 x + a 3 x 2 + a 4 x 3 + a 5 x 4 + a6 x 5 Therefore, N=20, M=6, M1 =7 To run the program, a result is produced as: A (1) =1. 77 546069 A(2)= -16 .62889 37 A(3) =11 5.42 272 A(4)=-394.29835 A(5)=6 57. 7 478 43 A(6)=- 416 . 677 009 HMAX=0.003 016 77 77 10 2 Chapter 4 A Development of Novel Integral Method Here, A (1) …A(6) are a1 …a6, and HMAX is the... (IM.EQ.IX (1) ) RETURN I =1 110 IF (IM.GE.IX(I)) THEN I=I +1 IF (I.LE.M1) GOTO 11 0 END IF IF (I.GT.M1) I=M1 IF (I.EQ.(I/2)*2) THEN H2=HH ELSE H2=-HH END IF IF (H1*H2.GE.0.0) THEN IX(I)=IM GOTO 20 END IF IF (IM.LT.IX (1) ) THEN DO 12 0 J=M ,1, -1 120 IX(J +1) =IX(J) IX (1) =IM GOTO 20 END IF IF (IM.GT.IX(M1)) THEN DO 13 0 J=2,M1 13 0 IX(J -1) =IX(J) IX(M1)=IM GOTO 20 END IF 10 5 10 6 Chapter 4 A Development of Novel Integral. .. WRITE(*,20)(I,A(I),I =1, M) WRITE(*,30)A(M1) 20 FORMAT (1X,'A(',I2,')=',D16.9) 30 FORMAT (1X,'HMAX=',D15.8) CLOSE (1) END SUBROUTINE HCHIR(X,Y,N,A,M,M1) DIMENSION X(N),Y(N),A(M1),IX(20),H(20) DOUBLE PRECISION X,Y,A,H,HA,HH,Y1,Y2,H1,H2,D,HM * * Appendix 4.A Fortran Program: Chebyshev Curve Fitting DO 5 I =1, M1 5 A(I)=0.0 IF(M.GE.N) M=N -1 IF(M.GE.20) M =19 M1=M +1 HA=0.0 IX (1) =1 IX(M1)=N L=(N -1) /M J=L DO 10 I=2,M IX(I)=J +1 J=J+L... IX(I)=J +1 J=J+L 10 CONTINUE 20 HH =1. 0 DO 30 I =1, M1 A(I)=Y(IX(I)) H(I)=-HH HH=-HH 30 CONTINUE DO 50 J =1, M II=M1 Y2=A(II) H2=H(II) DO 40 I=J,M D=X(IX(II))-X(IX(M1-I)) Y1=A(M-I+J) H1=H(M-I+J) A(II)=(Y2-Y1)/D H(II)=(H2-H1)/D II=M-I+J Y2=Y1 H2=H1 40 CONTINUE 10 3 10 4 50 Chapter 4 A Development of Novel Integral Method CONTINUE HH=-A(M1)/H(M1) DO 60 I =1, M1 60 A(I)=A(I)+H(I)*HH DO 80 J =1, M -1 II=M-J D=X(IX(II))... 0 .1, ξ (0) =0.5 α (0) = 0 .7, ξ (0) =0.5 α (0) = 0.9, ξ (0) =0.5 1. 8 1. 6 P02 _ P 01 1.4 1. 2 1. 0 0.8 0 10 20 30 40 θo 50 60 70 Fig 4 .16 The computed compressor total pressure ratio versus inlet flow angle 4.3 Results and Discussion α (0) = 0 .1, ξ (0) =0 .1 α (0) = 0 .7, ξ (0) =0 .1 α (0) = 0.9, ξ (0) =0 .1 α (0) = 0 .1, ξ (0) =0.5 α (0) = 0 .7, ξ (0) =0.5 α (0) = 0.9, ξ (0) =0.5 4.5 4.0 3.5 3.0 P2 -P1 97 2.5... Y2=A(II) DO 70 K=M1-J,M Y1=A(K) A(II)=Y2-D*Y1 Y2=Y1 II=K 70 80 CONTINUE CONTINUE HM=ABS(HH) IF (HM.LE.HA) THEN A(M1)=-HM RETURN END IF A(M1)=HM HA=HM IM=IX (1) H1=HH J =1 DO 10 0 I =1, N IF (I.EQ.IX(J)) THEN IF (J.LT.M1) J=J +1 ELSE H2=A(M) DO 90 K=M -1, 1, -1 90 H2=H2*X(I)+A(K) H2=H2-Y(I) IF (ABS(H2).GT.HM) THEN Appendix 4.A Fortran Program: Chebyshev Curve Fitting HM=ABS(H2) H1=H2 IM=I END IF END IF 10 0 CONTINUE... X(20),Y(20),A (7) DOUBLE PRECISION X,Y,A DATA X/0 .14 ,0 .16 ,0 .18 ,0.20, & 0.22,0.24,0.26,0.28,0.30,0.32,0.34, & 0.36,0.38,0.40,0. 415 ,0.42,0.44,0.46,0.48,0.49/ DATA Y/0.855,0.845,0.8 37, 0.832, & 0.830,0.830,0.832,0.8 37, 0.845,0.855,0.865, & 0.880,0.900,0.925,0.955,0.960,0.995 ,1. 040 ,1. 080 ,1. 10/ OPEN(UNIT =1, FILE='CURVE.DAT') N=20 M=6 M1 =7 CALL HCHIR(X,Y,N,A,M,M1) WRITE (1, 20)(I,A(I),I =1, M) WRITE (1, 30)A(M1) WRITE(*,20)(I,A(I),I =1, M)... = 25 Ο 0.0 θ Ο = 15 Ο 0 .1 0.2 0.3 0.4 0.5 α (0) 0.6 0 .7 0.8 0.9 1. 0 Fig 4 .12 The predicted outlet size of distorted region vs α ( 0 ) with higher inlet size of distorted region of 0.5 94 Chapter 4 A Development of Novel Integral Method 0.22 ξ (0) = 0 .1 0.20 ξ (10 ) 0 .18 0 .16 0 .14 θΟ = 5 0 .12 θ Ο = 15 Ο θ Ο = 25 Ο 0 .10 0.0 Ο 0 .1 0.2 0.3 0.4 0.5 α (0) 0.6 0 .7 0.8 0.9 1. 0 Fig 4 .13 The predicted outlet... Development of Novel Integral Method 0.62 0.60 P 2 -P 1 0. 57 θΟ = 5 Ο θ Ο = 15 Ο θ Ο = 25 Ο 0.55 0.52 0.50 0. 47 0.45 0.5 ξ (0) = 0.5 0.6 0 .7 0.8 0.9 1. 0 M Fig 4 .18 The simulated compressor characteristics at higher inlet size of distorted region of 0.5 0.62 0.60 P 2 -P 1 0. 57 θΟ = 5 Ο θ Ο = 15 Ο θ Ο = 25 Ο 0.55 0.52 0.50 0. 47 0.45 ξ (0) = 0 .1 0.92 0.94 0.96 0.98 1. 00 M Fig 4 .19 The simulated compressor characteristics... model for compressor stall cell propagation Transactions of the ASME, Journal of Engineering for Power, 10 4: 17 0 - 17 6 Emmons H.W., Pearson C.E and Grant H.P., 19 55, Compressor surge and stall propagation Transactions of the ASME, 77 : 455-469 Greitzer E.M., 19 80, Review: axial compressor stall phenomena ASME Journal of Fluids Engineering, 10 2: 13 4 -15 1 Jonnavithula S., Thangam S and Sisto F., 19 90, Computational . integrating (4 .1) along η-direction: ttanconsud 1 1 = ∫ − ηδ (4.9a) [] { } ∫∫∫ −−− = ∂ ∂ ′ + ′ −+ 1 1 x 1 1 1 1 2 dFd p )Y( dx d du dx d ηδη ρη ηδδηδ (4.9b) 11 11 ( , 1) ( , 1) y dpp uv d. contant 2R 11 000 11 Ud Ud π δ δα ηδ α η − − +≡ ∫∫ (4 .15 ) yields: 00000 K)RU2()]U)(2R2(U2[ ≡−+ παδπαδ (4 .16 ) Using (4 .10 ): 0 011 K)K1(K =−+ αα (4 . 17 ) Rearranged as: 1 10 0 K )KK( − − = α α α . l C 0 .10 62 0 .15 α =+ ( 810 α −°≤ ≤ ° ) (4.26a) 5 l 2 4 l 3 3 l 2 2 l 2 l 32 d C1044 918 .0C1020 416 .0C1060949.0 C1 079 495.0C1096 313 . 010 6 518 .0C −−− −−− ×+×−× −×+×−×= ( 1& amp; 2 α α < −° >